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Kyeong-Hun Kim

Publications and source records attributed to Kyeong-Hun Kim.

At least 37 records · Page 2Linked to original sources
arXiv2015

Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations

Let $p(t,x)$ be the fundamental solution to the problem $$ \partial_{t}^αu=-(-Δ)^βu, \quad α\in (0,2), \, β\in (0,\infty). $$ In this paper we provide the asymptotic behaviors and sharp upper bounds of $p(t,x)$ and its space and time fractional derivatives $$ D_{x}^{n}(-Δ_x)^γD_{t}^σI_{t}^δp(t,x), \quad \forall\,\, n\in\mathbb{Z}_{+}, \,\, γ\in[0,β],\,\, σ, δ\in[0,\infty), $$ where $D_{x}^n$ is a partial derivative of order $n$ with respect to $x$, $(-Δ_x)^γ$ is a fractional Laplace operator and $D_{t}^σ$ and $I_{t}^δ$ are Riemann-Liouville fractional derivative and integral respectively.

math.AP↗
arXiv2015

An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients

We introduce an $L_q(L_p)$-theory for the quasi-linear fractional equations of the type $$ \partial^α_t u(t,x)=a^{ij}(t,x)u_{x^i x^j}(t,x)+f(t,x,u), \quad t>0, \,x\in \mathbf{R}^d. $$ Here, $α\in (0,2)$, $p,q>1$, and $\partial^α_t$ is the Caupto fractional derivative of order $α$. Uniqueness, existence, and $L_q(L_p)$-estimates of solutions are obtained. The leading coefficients $a^{ij}(t,x)$ are assumed to be piecewise continuous in $t$ and uniformly continuous in $x$. In particular $a^{ij}(t,x)$ are allowed to be discontinuous with respect to the time variable. Our approach is based on classical tools in PDE theories such as the Marcinkiewicz interpolation theorem, the Calderon-Zygmund theorem, and perturbation arguments.

math.AP↗
arXiv2015

An $L_q(L_p)$-theory for parabolic pseudo-differential equations: Calderón-Zygmund approach

In this paper we present a Calderón-Zygmund approach for a large class of parabolic equations with pseudo-differential operators $\mathcal{A}(t)$ of arbitrary order $γ\in(0,\infty)$. It is assumed that $\cA(t)$ is merely measurable with respect to the time variable. The unique solvability of the equation $$ \frac{\partial u}{\partial t}=\cA u-λu+f, \quad (t,x)\in \fR^{d+1} $$ and the $L_{q}(\fR,L_{p})$-estimate $$ \|u_{t}\|_{L_{q}(\fR,L_{p})}+\|(-Δ)^{γ/2}u\|_{L_{q}(\fR,L_{p})} +λ\|u\|_{L_{q}(\fR,L_{p})}\leq N\|f\|_{L_{q}(\fR,L_{p})} $$ are obtained for any $λ> 0$ and $p,q\in (1,\infty)$.

math.AP↗
arXiv2014

Parabolic BMO estimates for pseudo-differential operators of arbitrary order

In this article we prove the BMO-$L_{\infty}$ estimate $$ \|(-Δ)^{γ/2} u\|_{BMO(\mathbf{R}^{d+1})}\leq N \|\frac{\partial}{\partial t}u-A(t)u\|_{L_{\infty}(\mathbf{R}^{d+1})}, \quad \forall\, u\in C^{\infty}_c(\mathbf{R}^{d+1}) $$ for a wide class of pseudo-differential operators $A(t)$ of order $γ\in (0,\infty)$. The coefficients of $A(t)$ are assumed to be merely measurable in time variable. As an application to the equation $$ \frac{\partial}{\partial t}u=A(t)u+f,\quad t\in \mathbf{R} $$ we prove that for any $u\in C^{\infty}_c(\mathbf{R}^{d+1})$ $$ \|u_t\|_{L_p(\mathbf{R}^{d+1})}+\|(-Δ)^{γ/2}u\|_{L_p(\mathbf{R}^{d+1})}\leq N\|u_t-A(t)u\|_{L_p(\mathbf{R}^{d+1})}, $$ where $p\ in (1,\infty)$ and the constant $N$ is independent of $u$.

math.AP↗
arXiv2014

Fractional time stochastic partial differential equations

In this paper, we introduce a class of stochastic partial differential equations (SPDEs) with fractional time-derivatives, and study the $L_2$-theory of the equations. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.

math.PR↗
arXiv2014

An L_p-theory for a class of non-local elliptic equations related to nonsymmetric measurable kernels

We study the integro-differential operators $L$ with kernels $K(y) = a(y) J(y)$, where $J(y)dy$ is a Lévy measure on $\bR^d$ (i.e. $\int_{\bR^d}(1\wedge |y|^2)J(y)dy<\infty$) and $a(y)$ is an only measurable function with positive lower and upper bounds. Under few additional conditions on $J(y)$, we prove the unique solvability of the equation $Lu-λu=f$ in $L_p$-spaces and present some $L_p$-estimates of the solutions.

math.AP↗
arXiv2013

On the Lq(Lp)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $O \subset R^d$ with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces $\mathfrak{H}^{γ,q}_{p,θ}(O;T)$. The summability parameters p and q in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the Hölder regularity in time is analysed. Moreover, we prove a general embedding of weighted Lp(O)-Sobolev spaces into the scale of Besov spaces $B^α_{τ,τ}(O), 1/τ=α/d+1/p, α> 0$. This leads to a Hölder-Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.

math.PR↗
arXiv2012

A weighted $L_p$-theory for parabolic PDEs with BMO coefficients on $C^1$-domains

In this paper we present a weighted $L_p$-theory of second-order parabolic partial differential equations defined on $C^1$ domains. The leading coefficients are assumed to be measurable in time variable and have VMO (vanishing mean oscillation) or small BMO (bounded mean oscillation) with respect to space variables, and lower order coefficients are allowed to be unbounded and to blow up near the boundary. Our BMO condition is slightly relaxed than the others in the literature.

math.AP↗
arXiv2011

A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth domains

In this paper we study parabolic stochastic partial differential equations defined on arbitrary bounded domain $\cO \subset \bR^d$ allowing Hardy inequality: $$ \int_{\cO}|ρ^{-1}g|^2\,dx\leq C\int_{\cO}|g_x|^2 dx, \quad \forall g\in C^{\infty}_0(\cO), $$ where $ρ(x)=\text{dist}(x,\partial \cO)$. Existence and uniqueness results are given in weighted Sobolev spaces $\frH^γ_{p,θ}(\cO,T)$, where $p\in [2,\infty)$, $γ\in \bR$ is the number of derivatives of solutions and $θ$ controls the boundary behavior of solutions. Furthermore several Hölder estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.

math.PR↗
arXiv2011

A $W^n_2$-Theory of Stochastic Parabolic Partial Differential Systems on $C^1$-domains

In this article we present a $W^n_2$-theory of stochastic parabolic partial differential systems. In particular, we focus on non-divergent type. The space domains we consider are $\bR^d$, $\bR^d_+$ and eventually general bounded $C^1$-domains $\mathcal{O}$. By the nature of stochastic parabolic equations we need weighted Sobolev spaces to prove the existence and the uniqueness. In our choice of spaces we allow the derivatives of the solution to blow up near the boundary and moreover the coefficients of the systems are allowed to oscillate to a great extent or blow up near the boundary.

math.PR↗
arXiv2010

A $W^n_2$-Theory of Elliptic and Parabolic Partial Differential Systems in $C^1$ domains

In this paper second-order elliptic and parabolic partial differential systems are considered on $C^1$ domains. Existence and uniqueness results are obtained in terms of Sobolev spaces with weights so that we allow the derivatives of the solutions to blow up near the boundary. The coefficients of the systems are allowed to substantially oscillate or blow up near the boundary.

math.AP↗
arXiv2010

A $W^1_2$-theory of Stochastic Partial Differential Systems of Divergence type on $C^1$ domains

In this paper we study the stochastic partial differential systems of divergence type with $C^1$ space domains in $\bR^d$. Existence and uniqueness results are obtained in terms of Sobolev spaces with weights so that we allow the derivatives of the solution to blow up near the boundary. The coefficients of the systems are only measurable and are allowed to blow up near the boundary.

math.PR↗
arXiv2010

An $L^2$-theory on SPDE driven by Lévy processes

In this paper we develop an $L_2$-theory for stochastic partial differential equations driven by Lévy processes. The coefficients of the equations are random functions depending on time and space variables, and no smoothness assumption of the coefficients is assumed.

math.PR↗
arXiv2010

An $L^p$-theory of non-divergence form SPDEs driven by Lévy processes

In this paper we present an $L^p$-theory for the stochastic partial differential equations (SPDEs in abbreciation) driven by Lé{}vy processes. Existence and uniqueness of solutions in Sobolev spaces are obtained. The coefficients of SPDEs under consideration are random functions depending on time and space variables.

math.PR↗